Analytic Geometry Answers

1. The sides of a triangle are given by the equations 2x + y = 5, x + 2y = 7 and
x - y = 1 find the vertices of the triangle and illustrate with a sketch.

2. Show that the three lines given by the equations 3x + 5y + 7 = 0, x + 2y + 2 = 0 and 2x - y + 9 =0 are concurrent (i. e. Pass through the one point)

1. Find the point of intersection of the following pairs of linea whose equations are given.
a) x + 3y = 9 and 5x - 2y = 11
b) 4x + 3y =8 and 6x - 2y = - 14
c) 3x + 2y - 7 =0 and 5x - 6y = 7

2. Find the equation of the straight line which passes through the origin and through the point of intersection of the lines 4x - y- 3 =0 and x + 2y - 12=0

Find the distance of the centre of the circle x^2+y^2+z^2+x-2y+2z=3, 2x+y+2z=1 from the plane ax+by+cz=d where a,b,c,d are constants and also find the equation of the right circular cylinder whose base curve is the circle obtained above.
above.

Can any conic have its focus lying on the corresponding directrix? Give reasons for
your answer.

Identify an axis of revolution and generating conic of the surface 4x^2+25y^2+4z^2=100. Does this conic also generate x^2/4 + y^2/25 + z^2/4=1?Give reasons for your answer.

A rotating liquid forms a surface in the form of a paraboloid. The surface is 2m
deep at the centre and 10m across. Obtain an equation of the surface.

Find the equation of the conic of which one focus lies at (2,1) one directrix is
x + y = 0 and it passes through (1,4) Also identify the conic and reduce the conic you obtained above to standard form.
Draw a rough sketch of the conic obtained above.

Every conicoid has a unique centre.
Is the statement true? Give reason for your answers either with a
short proof or a counter example

ax^2+by^2+cz^2=d represents a sphere with radius √(a^2+b^2+c^2-d) where
a,b,c,d are positive real numbers.
Is the statement true? Give reason for your answers either with a
short proof or a counterexample.

Given any two conics C1 and C2 , at least one element of R× R will belong to
C1 ∩C2.
Is the statement true? Give reason for your answers either with a
short proof or a counterexample.